2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/208606We prove localization and Zariski-Mayer-Vietoris for higher Grothendieck-Witt groups, alias hermitian $K$-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be singular. Along the way, we prove additivity, fibration and approximation theorems for the hermitian $K$-theory of exact categories with weak equivalences and duality.K-Theory and HomologyAlgebraic Geometry19G38, 19E08, 19G12, 11E81, 14C35text